Published in the Proceedings of the International Symposium on the Foundations of Modern Physics 1993, Helsinki, Finland, ed. Hyvonen, T., World Scientic, Singapore, 35 - 56. Quantum Probability, the Classical Limit and Non-Locality. Diederik Aerts, Thomas Durt and Bruno Van Bogaert, FUND, Free University of Brussels, Pleinlaan 2, B-1050 Brussels, Belgium, [email protected] ABSTRACT. We investigate quantum mechanics using an approach where the quantum probabilities arise as a consequence of the presence of uc- tuations on the experimental apparatuses. We show that the full quantum structure can be obtained in this way and dene the classi- cal limit as the physical situation that arises when the uctuations on the experimental apparatuses disappear. In the limit case we come to a classical structure but in between we nd structures that are nei- ther quantum nor classical. In this sense, our approach not only gives an explanation for the non-classical structure of quantum theory, but also makes it possible to dene, and study the structure describing the intermediate new situations. By investigating in which way the non-local quantum behaviour disappears during the limiting process we can explain the ’apparent’ locality of the classical macroscopical world. 1. Introduction. Many aspects of quantum mechanics remain puzzling, still today, more than sixty years after its birth. We want to concentrate on three of these paradoxical aspects, and explain in which way the approach that we have elaborated in our Brussels group, can provide an answer to them. 1.1 The Origin of the Quantum Probabilities. The probabilistic behaviour of quantum entities has been of great concern to the community of physicists working on the foundations of the theory. Indeterminism in itself is not a problem, after all also statistical mechanics is a non-deterministic theory. The structure of the quantum probabilities however seems to make it im- possible to construct an underlying, even hypothetical, deterministic theory. This indicates that quantum indeterminism does not nd its origin in a lack of knowl- edge about a deeper objective reality, as it is the case for the probabilistic structure 1 of classical statistical mechanics. As a consequence, the quantum probabilities are often considered as ’intrinsically present in nature’, and whatever this means, if it is true, it makes the existence of an ’objective’ reality problematic. Whether quantum indeterminism originates in a lack of knowledge about a deeper, still unknown, reality of the entity under study, has been investigated in great detail. A possible ’deterministic’ generalization of a probabilistic theory in this sense has been called a ’hidden variable’ theory. Already during the rst years of quantum theory, John Von Neumann formulated a ’no-go-theorem’ 1 about hidden-variable theories for quantum mechanics. Von Neumann’s original no-go-theorem was crit- icised by several authors 2, and then rened by others 3, leading nally to the following conclusion : when one forces a ’mathematical’ hidden variable theory, recovering all the predictions of quantum mechanics, it always contains an aspect that makes it very un-physical, namely, the hidden variables dening the micro- state depend explicitly on the experiment that one considers. In this sense, this kind of hidden variables, ’contextual’ as they have been called, cannot be given a physical meaning when attached to the physical entity under study, and therefore they were considered to be of no physical interest. With these results many of the mathematically oriented physicists were convinced that ’physical’ hidden variables theories were impossible for quantum mechanics. John Von Neumann wrote down his no-go-theorem in 1932, and three years later Einstein, Podolsky and Rosen presented their famous paper 4, where they show that quantum mechanics is an incomplete theory. EPR proof that, if quantum mechanics can be correctly applied to a situation of separated physical entities, it is possible to dene elements of re- ality of the states of the entities, not contained in the quantum description. This means that there do exists hidden variables in the sense of Von Neumann. David Bohm presented another version of the EPR reasoning, and constructed explicitly a hidden variable theory for quantum mechanics. Also John Bell was convinced by the result of the EPR paper, the incompleteness of quantum mechanics, and the presence of hidden variables to complete it. He derived his famous inequalities, that should be violated if nature follows quantum mechanics in its description of entities detectable in widely separated regions of space. This was the situation at the end of the sixties : Von Neumann’s theorem and its renements giving rise to the belief in the impossibility of hidden variable theories, and the EPR reasoning and its elaborations, indicating the existence of hidden variables com- pleting quantum mechanics. In our Brussels group, we have developed an approach 7,8,9,10,11,12,13, where the quantum probabilities are derived as a consequence of the presence of uctuations in the experimental apparatuses. The quantum state 2 is considered to be a pure state, hence there are no hidden variables in the sense of Von Neumann, describing a deeper reality of the entity under study, but the experiments are not pure. Dierent experiments, considered to be equivalent on the macroscopical level, and hence presented by the same mathematical concept in the theory, are not identical on a deeper level. This lack of knowledge about the deeper reality of the experimental apparatuses and their interaction with the entity, is in our approach the origin of the presence of the quantum probabilities. We can easily understand why our approach allows us to construct ’classical’ mod- els giving rise to quantum probability structures, or with other words, why our examples escape the no-go-theorems about hidden variable theories. Indeed, on a purely mathematical level, we can describe the uctuations in the experimental apparatuses by means of a variable, and interpret this as a hidden variable. The mathematical model then reproduces, as an hidden variable theory, the quantum probabilities. But, since the variable belongs to the experimental apparatus, and not to the state of the entity, this hidden variable theory is highly contextual, and hence escapes the no-go-theorems. 1.2 The Classical Limit Although there are connections on dierent levels between quantum mechanics and classical mechanics, the fundamental relation between both theories is still obscure. Many general axiomatic approaches lead to structures that incorporate quantum as well as classical theories, but also here the fundamental nature of the relation is not understood. Let us indicate the situation connected with the study of the relation between quantum and classical by the problem of the ’classical limit’. In our approach there is a straightforward way to investigate the relation between quantum mechanics and classical mechanics. Indeed, if the quantum probabilities nd their origin in the presence of uctuations on the experimental apparatuses, we consider the quantum situation as the one where these uctuations are maximal, the classical situation as the one where they are zero, and in between we have a new type of situation, that we call ’intermediate’. These ’intermediate’ situations give rise to a structure, for the collection of properties and for the probability model, that is neither quantum nor classical. In our approach we have parametrized the amount of uctuations by means of a number ∈ [0, 1], where = 0 corresponds to the classical situation of no uctuations, leading to classical structures (Boolean lattice of properties and Kolmogorovian probability model), = 1 to the quantum situation of maximal uctuations, leading to quantum structures (quantum lattice of properties and quantum probability model), and 0 <<1 to intermediate 3 situations, giving rise to a property-lattice and a probability model that are neither Boolean or Kolmogorovian nor quantum. In 12,13 we study explicitly a physical example, that we have called the -example, and that gives rise to a model that 1 is isomorphic with the quantum mechanical description of the spin of a spin 2 quantum entity for the case =1.For= 0 the -example describes a classical mechanics situation, and for values of between 0 and 1 we nd intermediate situations, with structures that are neither quantum nor classical, as is shown explicitly in 12,13. The limit → 0 is the classical limit in our approach. We have generalized this approach to the realistic situation of a quantum entity described in an innite dimensional Hilbert space L2(<3), and again we dene a continuous limit process, → 0, corresponding to a continuous decrease of uctuations on the measurement apparatuses. Starting with a presentation of the two dimensional case in section 2, we investigate the n-dimensional case in section 3, and the general, realistic and innite dimensional case, in section 4. 1.3 Non-Locality. The theoretical prediction and experimental verication of non-local quantum be- haviour, in the form of non-local correlations (the experiments to verify Bell’s inequalities), and as a manifestly present reality (Rauch’s experiments with the neutron-interferometer), is a more recent, but as dicult to understand aspect of quantum theory. Many physicist have pointed out many problems caused by the presence of non-locality, only to mention : ’causality’, and ’classical separability’ of distant entities. The investigations of Bell’s inequalities have lead to experiments on quantum entities that one tries to keep in a quantum mechanical superposition state (non-product state), while they are detected in largely separated regions of space. These experiments have shown that the quantum non-separation of the two entities, giving rise to a type of correlations that have been called quantum cor- relations, remains intact, even under these extreme experimental situations 14,15. Experiments of the group of Helmut Rauch with ultra-cold neutrons and a neu- tron interferometer have confronted us in a more direct way with this non-local behaviour of quantum entities 16.